Main (and initial) synthesis goal is to recreate natural sounds from artificial and basics waveforms. Today, with new synth techiques and new technologies, this goal was widely expanded and, now, synthesis is a way to create powerful new sounds (both emulation of acustic ones and totally new flavours).

Subtractive Synthesis
One of first and, even today, evergreen and powerful synth technique was subctrative synthesis. To let you understand better, you have to know that a sound is made by a fundamental (that our brain recognise as tone.. i.e.: C or D) and harmonics (that gives a sound its timbre.. a C played by a flute sounds different than a C played by a piano). Subtracive synthesis uses Filters to subtract harmonic frequencies by a starting point, called usually CutOff and let pass other frequences depending on which kind of filter you are using. A Lowpass filter let pass frequencies below CutOff point, while Highpass filters does the opposite so frequencies higher than CutOff passes through and so on...

Analog What?
Firts synthesizers was made totally analogic.. this means that, inside of them, you can't find chips/DSPs and so on... but only resistors, condenser etc.. today most synthesizers are totally digital so, instead of resistors etc, there are a main DSP that recreate those circuitations via software.
Why Analogic, today, is ever appreciated? This is because analogue circuirties aren't perfect and those imperfections made their sound different from digital ones (that are mathematically exact)... but today we can even look at Virtual Analog.. means that a digital boards are programmed to reproduce those imperfections of old analog boards.. I wanna give you a classical example.. pick an old Minimoog Model D and a modern Kurzweil K2*00.. with analog filters you can make them to "auto oscillate" simply rising their resonance to highest values.. this is not possible on digital filters (but it is possible with Virtual Analog devices because they are programmed to simulate old boards attitude).

About VAST
Kurzweil VAST stands for Variable Architecture Synthesis Technology. Kurzweil claims that their synths can do a lot of synths techinique with this platform... well.. this is almost true. It uses algorithms composed by some blocks.. each algorithm have different blocks, displaced in a different way and different type of modules you can use.

Most of modern synths have only one algorithms.. look at their manual and verify:

Code:

[Oscillator]-->[Filter]-->[Amp]

This is the absolute basic and common algorithm you may find in every board you can own.. from old Moogs to newst Korg Oasys. Some of them have little differencies that makes a synth more complex/versatile then others.. but, IMHO, there are 2 things that really makes the difference between synths:

1) Their filters quality
2) Their Modulation Matrix

How to test filters?
There are an easy way to compare filters. Lets look at common and widely used Lowpass filter. Make a patch that uses a White Noise as oscillator and pass it through a lowpass. Set its cutoff to lowest frequencies and assign your Modulation Wheel (or whatever controller you have) to CutOff in a manner that you can open/close the filter for all its frequency range. I have to find a page where a guy posted a Moog Voyager sample of that.. is a good way to compare digital filters with analog ones.

What is Modulation Matrix
Mod Matrix is a simple matrix (like an Excel Spreadsheet) where you have control sources on rows and destinations on columns. In this way you can know how many thinks can be modified and by what. Imagine.. you wanna make a pad with the filter slowly opens from a cutoff point to another and, when you move your expression pedal, a vibrato comes out..

Subtractive synthesis pretty much forms the basis of every major type of synthesis these days. Basically you start with a solid tone (ie waveform, such as a saw wave or a aquare wave). Then you "carve away" elements of that tone to get something more "lively" or interesting. You can do this with filters or envelope generators, or filters controlled by envelope generators and so forth. You can also generally "modulate" aspects of the sound by different means such as a low frequency oscillator (which generally oscillates at a frequency below audio range, hence the name), or some type of real time controller like a modulation wheel. You can get quite a lot of different sounds using these few methods, depending on the capabilities of the synth itself and it's voice architecture (how the different sound altering elements are linked together, which is sometimes variable).

Analog just means the sythesis takes place in the electrical realm, using resitors, transistors, capacitors, switches, integrated circuits (like the old Curtis chips) and other basic electrical components. Whereas digital sythesis basically uses small function dedicated computers to carry out these sound alterations. Both methods have their advantages and disadvantages. For instance, digital tends to be more stable, but analog has a tremendous sound.